Bounds for the effective coefficients of homogenized low dimensional structures

For a given amount $m$ of mass, we study the class of
materials which can be reached by homogenization distributing the
mass $m$ on periodic structures of prescribed dimension $k\leq n$
in $\ren$. Both in the scalar case of conductivity and in the
vector-valued case of elasticity, we find some bounds for the
effective coefficients, depending on the mass $m$ and the
dimension parameters $k, n$. In the scalar case we prove that such
bounds are optimal, as they do describe the set of all materials
reachable by homogenization of structures of the type under
consideration; in the vector-valued case we show that some of our
estimates are attained.